Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 12.8

Time: 48.4s

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.673221847358114621753000328052530955067 \cdot 10^{-199} \lor \neg \left(t \le 9.430149747819988364651382548473002399861 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\frac{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}{\frac{2}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{k \cdot t}{\ell} - \frac{1}{6} \cdot \frac{{k}^{3} \cdot t}{\ell}\right)\right) \cdot \sin k}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r121347 = 2.0;
        double r121348 = t;
        double r121349 = 3.0;
        double r121350 = pow(r121348, r121349);
        double r121351 = l;
        double r121352 = r121351 * r121351;
        double r121353 = r121350 / r121352;
        double r121354 = k;
        double r121355 = sin(r121354);
        double r121356 = r121353 * r121355;
        double r121357 = tan(r121354);
        double r121358 = r121356 * r121357;
        double r121359 = 1.0;
        double r121360 = r121354 / r121348;
        double r121361 = pow(r121360, r121347);
        double r121362 = r121359 + r121361;
        double r121363 = r121362 + r121359;
        double r121364 = r121358 * r121363;
        double r121365 = r121347 / r121364;
        return r121365;
}

↓

double f(double t, double l, double k) {
        double r121366 = t;
        double r121367 = -2.6732218473581146e-199;
        bool r121368 = r121366 <= r121367;
        double r121369 = 9.430149747819988e-188;
        bool r121370 = r121366 <= r121369;
        double r121371 = !r121370;
        bool r121372 = r121368 || r121371;
        double r121373 = 1.0;
        double r121374 = cbrt(r121366);
        double r121375 = 3.0;
        double r121376 = pow(r121374, r121375);
        double r121377 = l;
        double r121378 = r121376 / r121377;
        double r121379 = k;
        double r121380 = sin(r121379);
        double r121381 = r121378 * r121380;
        double r121382 = r121376 * r121381;
        double r121383 = r121373 / r121382;
        double r121384 = 2.0;
        double r121385 = 1.0;
        double r121386 = r121379 / r121366;
        double r121387 = 2.0;
        double r121388 = pow(r121386, r121387);
        double r121389 = fma(r121384, r121385, r121388);
        double r121390 = r121377 / r121376;
        double r121391 = cos(r121379);
        double r121392 = r121390 * r121391;
        double r121393 = r121389 / r121392;
        double r121394 = r121387 / r121380;
        double r121395 = r121393 / r121394;
        double r121396 = r121383 / r121395;
        double r121397 = r121379 * r121366;
        double r121398 = r121397 / r121377;
        double r121399 = 0.16666666666666666;
        double r121400 = 3.0;
        double r121401 = pow(r121379, r121400);
        double r121402 = r121401 * r121366;
        double r121403 = r121402 / r121377;
        double r121404 = r121399 * r121403;
        double r121405 = r121398 - r121404;
        double r121406 = r121376 * r121405;
        double r121407 = r121406 * r121380;
        double r121408 = r121387 / r121407;
        double r121409 = r121408 / r121393;
        double r121410 = r121372 ? r121396 : r121409;
        return r121410;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -2.6732218473581146e-199 or 9.430149747819988e-188 < t

   1. Initial program 27.8

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 27.8

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 28.0

      \[\leadsto \frac{\frac{2}{\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   5. Applied unpow-prod-down 28.0

      \[\leadsto \frac{\frac{2}{\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   6. Applied times-frac 19.8

      \[\leadsto \frac{\frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   7. Applied associate-*l* 17.9

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied unpow-prod-down 17.9

      \[\leadsto \frac{\frac{2}{\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied associate-/l* 13.0

       \[\leadsto \frac{\frac{2}{\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   11. Using strategy rm

   12. Applied tan-quot 13.0

       \[\leadsto \frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   13. Applied associate-*l/ 12.2

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{\sin k}{\cos k}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied frac-times 11.0

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   15. Applied associate-/r/ 11.0

       \[\leadsto \frac{\color{blue}{\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k} \cdot \left(\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   16. Applied associate-/l* 10.0

       \[\leadsto \color{blue}{\frac{\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}}\]
   17. Using strategy rm

   18. Applied *-un-lft-identity 10.0

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\]

   19. Applied times-frac 10.0

       \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)} \cdot \frac{2}{\sin k}}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\]

   20. Applied associate-/l* 8.3

       \[\leadsto \color{blue}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\frac{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}{\frac{2}{\sin k}}}}\]

   if -2.6732218473581146e-199 < t < 9.430149747819988e-188

   1. Initial program 64.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 64.0

      \[\leadsto \frac{\frac{2}{\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   5. Applied unpow-prod-down 64.0

      \[\leadsto \frac{\frac{2}{\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   6. Applied times-frac 64.0

      \[\leadsto \frac{\frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   7. Applied associate-*l* 64.0

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied unpow-prod-down 64.0

      \[\leadsto \frac{\frac{2}{\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied associate-/l* 53.7

       \[\leadsto \frac{\frac{2}{\left(\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   11. Using strategy rm

   12. Applied tan-quot 53.7

       \[\leadsto \frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   13. Applied associate-*l/ 53.7

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{\sin k}{\cos k}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied frac-times 54.0

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   15. Applied associate-/r/ 54.1

       \[\leadsto \frac{\color{blue}{\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k} \cdot \left(\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   16. Applied associate-/l* 52.3

       \[\leadsto \color{blue}{\frac{\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \sin k}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}}\]

   17. Taylor expanded around 0 43.7

       \[\leadsto \frac{\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \color{blue}{\left(\frac{k \cdot t}{\ell} - \frac{1}{6} \cdot \frac{{k}^{3} \cdot t}{\ell}\right)}\right) \cdot \sin k}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.673221847358114621753000328052530955067 \cdot 10^{-199} \lor \neg \left(t \le 9.430149747819988364651382548473002399861 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\frac{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}{\frac{2}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{k \cdot t}{\ell} - \frac{1}{6} \cdot \frac{{k}^{3} \cdot t}{\ell}\right)\right) \cdot \sin k}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))